Have you ever heard about black smokers? They are hydrothermal vents on the ocean floor, hosting a very peculiar biodiversity.
Here, we can find the copper-shelled scaly-foot snail as suggested by Pelinore, so it is a good starting point to imagine a lifeform with a copper skeleton.
The rationale we are using calcium to build bony structure is that it can be assembled with oxygen to form hydroxyapatite, a high-resistance crystal which, one stockpiled in an organic matrix made of regular proteins, grants exceptional solidity. Note that the calcium itself is not bound to the organic molecules, see detailed paper on how it looks here.
Calcium is a very good material because of its relative abundancy (unlike titane or tungsten), ease to assemble/disassemble in crystal (unlike copper) and low interference with other organic processes (unlike lead or cadmium). But let's consider your creature is using copper.
Hydrothermal vents likely exists on other bodies in the solar system and beyond, such as Enceladus or Europa. Here, without interference from a sunlight-powered ecosystem and provided that the vents last long enough for evolution to work, we can imagine exotic creatures using copper released from the planet's core for their metabolism.
You may be interested to learn that blue-purple copper-based hemoglobin exist in a variety of animals on Earth and could be the predominant color of your creature.
As for the bones, metallic copper is not really hard and instead its oxydes could be used. Copper oxyde(I) is red while copper oxyde(II) is dark-blueish, pick as you like.
Finally, copper is a good electron carrier and one can imagine that a copper-based creature would have electrical properties, essential for sensing in an otherwise obscure environment.